Basic operations on complex numbers, how to calculate complex numbers

This uses Euler's formula.

cosa+i*sina=e^(ia)

e is the base of the natural logarithm, e=

cosa+i*sina) to the nth power is e (ina), which is cos(na) + i*sin(na).

I 2011 is converted to the form of a polar coordinate, i.e., [1,90 degrees*2011] = > [1,270 degrees].

Convert i2 to the form of the polar coordinates, i.e., [ 5, ] where tan = -2 and. for the 4th quadrant angle).

i^2011

i2) = [ 5 5,270 degrees - ].

i^2011

i2)=[ 5 5, -270 degrees]=[ 5 5, ] where tan is 1 2, is the 1st quadrant angle)

i^2011

i2) for -i 2011

i2) point is shifted one unit to the right, so it is still in the first quadrant.

The operation of negative numbers includes the law of addition, the rule of multiplication, the rule of division, the rule of open, the law of arithmetic, the rule of multiplication of i, etc. Here's how to do it:

1.The law of addition.

The addition rule of complex numbers: let z1=a+bi, z2=c+di be any two complex numbers. The real part of the sum of the two is the sum of the original two complex real parts, and its imaginary part is the sum of the original two imaginary parts. The sum of two complex numbers is still plural. Namely.

2.The law of multiplication.

Multiplication rule of complex numbers: multiply two complex numbers, similar to multiplying two polynomials, where i2 = -1, merge the real part and the imaginary part separately. The product of two complex numbers is still a complex number. Namely.

3.The law of division.

Complex Division Definition: Satisfied.

Complex number of . <>

It is called the quotient of the plural number A + Bi divided by the plural number C + Di.

Operation: Multiply the numerator and denominator by the conjugate complex of the denominator at the same time, and then use the multiplication rule, i.e.

4.Prescribing the rules.

If zn=r(cos +isin), then.

（k=0，1，2，3?n-1）

5.Arithmetic.

Commutative law of addition: z1+z2=z2+z1

The commutative law of multiplication: z1 z2=z2 z1

Additive associativity: (z1+z2)+z3=z1+(z2+z3).

Multiplicative associativity: (z1 z2) z3=z1 (z2 z3).

Distributive property: z1 (z2+z3)=z1 z2+z1 z3

of multiplication.

i4n+1=i, i4n+2=-1, i4n+3=-i, i4n=1 (where n z).

7.DeMorfer's theorem.

For the complex number z=r(cos +isin), there is an nth power of z.

zn=rn[cos(n)+isin(n)], where n is a positive integer).

<> rule. <>

<> conjugate plural definition.

For complex numbers. <>

Call it a plural.

=a-bi is the conjugate complex of z. That is, the two real parts are equal, and the imaginary parts are opposite to each other, and the complex number is conjugate complex number. The conjugate plural of the complex number z is denoted as.

Quality. By definition, if.

(a,b r), then.

=a－bi（a,b∈r）。The points corresponding to the conjugate complex numbers are symmetrical with respect to the real axis. Two complex numbers: x+yi and x-yi are called conjugate complex numbers, their real parts are equal, and the imaginary parts are opposite to each other.

In the complex plane, the point representing two conjugate complex numbers is symmetrical with respect to the x-axis, and this is exactly what it is"Conjugation"The word ** --- two oxen pulling a plough in parallel, and on their shoulders they have to put a beam together, and this beam is called"Yokes"。If z is used to represent x+yi, then add a word z"one"It means x-yi, or vice versa.

Conjugate plurality has some interesting properties:

Ask the second question.

Write me the steps of the operation or something.

Answer Question 1 1 i

Question 2 2i-1

Question 3 21 x 20i